III 1 Theoretical Seismology Wave Propagation by j7EkoN

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									         Theoretical Seismology 2: Wave Propagation




Thailand Training Program in Seismology and Tsunami Warnings, May 2006
Theoretical Seismology 2: Wave Propagation

       Seismic waves in an elastic medium

       ・ Rays and Ray Paths
          How they propagate in the Earth
              Travel-time curves
       ・ Near-Field Terms (Static Displacements)
       • Far-Field Terms (P, S, Surface waves)
       • Surface Waves
       ・ Normal modes
          (Free oscillations of the Earth)
  Explosion




P waves
Psychology test

      Homogeneous Earth


                          (seismologist)




                                     (engineer)




                                           (tsunamicist?)
Ray Paths in a Layered Medium

Snell’s law:    sin a1 / sin a2 = a1 / a2



        a1     q1        Slower       a1    q1        Faster

                    q2
                             Faster              q2    Slower
      a2                              a2


               a1 < a2                      a1 > a2
Travel-time Curves for                        Travel-time curves
Ray Paths in a Layered Medium
                                Time

                                                        1/a3

                                              1/a2


                                       1/a1

                                         Distance
                                         (D=T/velocity)
a1      head wave


a2


a3
Velocity structure of the Earth
Ray Paths in a Gradient                                Travel-time curves

                                         Time
 Velocity gradient can be treated as a
 series of thin homogeneous layers.                                 1/a3

                                                       1/a2


                                                1/a1

                                                  Distance
                                                  (D=T/velocity)
a1


a2


a3
Moho

       Andrija Mohorovicic (1857-1936)

       Found seismic discontinuity at
       30 km depth in the Kupa Valley
       (Croatia).




       Mohorovicic discontinuity or ‘Moho’

       Boundary between crust and mantle
Structure in the Earth




                         Conrad and Moho Discontinuities




                              Low velocity zone
         Forward Branch




Receding Branch
         Forward Branch




       Shadow Zone


  Forward Branch (PKPbc)

Receding Branch (PKPab)
PcP
                                                            Receding
                                                            Branch     A
                Forward                                  PKP
                Branch                                      B    C
                                                             Forward
                                                             Branch

              Shadow
                                   PcP                 Shadow
              Zone                                 P   Zone
        Forward Branch

      Receding Branch                    Forward
                                         Branch




 Not shown: PKP(DEF) and PKiKP

 Other notation for core phases:
 ABC branch known as P2l
 DEF branch known as P1ll
 PKP(DEF) known as PKIKP

 Point B is a caustic
   PcP




Core Reflections
  Faulting




Seismic waves
Other aspects of wave propagation:
   • Diffracted Waves
  ・ Surface Waves
  ・ Static Displacements
  ・ Frequency content
   • Normal Modes
Other aspects of wave propagation
   • Diffracted Waves
  ・ Surface Waves
  ・ Static Displacements
  ・ Frequency content and wavelength
   • Normal Modes
1-D Wave Equation



    u1 1  u1
     2          2
        
   x12
          c t 2




1-D wave equation


c = propagation speed
 2 u1 1  2 u1
       
x1  2
         c t 2

Solution
     u( x, t )  A sin[ (t  x / c)]
              2       T = wave period
        T
                       = angular frequency




                                               LW 3.2.1
        Wave Period and Wavelength

                     Velocity = Wavelength / Period
Space            x
                           Velocity 6 km/s

        wavelength
                           period 50 s
                           Wavelength 300 km
Time
                 t
                        period 50 s
                        frequency = 1/period= 0.02 hz
        period
                       Period         Wavelength
Body waves          0.1 to 50 sec     50 m to 500 km
(P・S)

Surface waves       10 to 350 sec     30 to 1000 km


Free Oscillations   350 to 3600 sec   1000 to 10000 km


Static
Displacements
                                           -
Other aspects of wave propagation
   •Diffracted waves
  ・ Surface waves
  ・ Static Displacements
           (amplitude at zero frequency)
 ・ Frequency content
  • Normal modes
              3-D Wave Equation with Source

           2u
          2  f  (  2 )(  u )    (  u )
          t
                        source                         spatial 2nd derivative




                                                      Near-field Terms (Static Displacements)
Solution
          1        1      r/                           1             1          r    1        1         r
u ( x, t )             a       M 0 (t   )d             A IP      M 0 (t  )      A IS 2 M 0 (t  )
                 N
                 A 4
             4   r     r/                           4a 2          r2         a 4 2     r          

                 1             1        r    1       1       r
                       A FP     M 0 (t  )      A FS M 0 (t  )
               4a 3          r         a 4 3     r        


                                          Far-field Terms (P, S Waves)
Near-field terms



    ・ Static displacements
                                            r/a    r/
    ・ Only significant close to the fault

    ・ Source of tsunamis


                                             r/a r/

                                                  t →
Static Displacements




  Bei-Fung Bridge near Fung-Yan city, 1999 Chi-Chi, Taiwan earthquake
Static displacements



  Co-seismic deformation
  of 2003 Tokachi-oki
  Earthquake (M8.0)
     Generation of Tsunami from Near-field Term




EA
               PAC
Far-field Terms      
                           1
                                    A FP
                                           1        r
                                             M 0 (t  ) 
                                                          1       1       r
                                                              A FS M 0 (t  )
                         4a   3
                                           r         a 4 3
                                                                  r        




     ・ Propagating Waves

     ・ No net displacement
     in an elastic medium

     ・ P waves



     ・ S waves
Other aspects of wave propagation
   • Diffracted Waves
  ・ Surface Waves
  ・ Static Displacements
  ・ Frequency content
   • Normal Modes
    Surface Waves




                        Group Velocity (km/sec)
                                                    Love




                                                        Rayleigh



S                                                 Period (sec)



    Shearer, Fig. 8.1
January 26, 2001 Gujarat, India Earthquake (Mw7.7)


       Body waves

                                              vertical
                           Rayleigh Waves

          P    PP   S      SS



                                               radial




                                              transverse
                        Love Waves




   Recorded in Japan at a distance of 57o (6300 km)
Other aspects of wave propagation
   • Diffracted Waves
  ・ Surface Waves
  ・ Static Displacements
  ・ Frequency content
   • Normal Modes
                 Free Oscillations of the Earth
                       (Normal Modes)




    Few minutes after the earthquake                 Few hours after the earthquake (0S20)
    Constructive interferences  free oscillations
    (or stationary waves)

 Standing Waves with Periods < 54 min, amplitudes < 1 mm

 Observable months after great earthquakes (e.g. Sumatra, Dec 2004)


                                                     From Michel van Camp, Royal Obs. of Belgium
    Normal Modes




         (Stein and Gellar 1978)



Free Oscillations of the Earth     (Daishinji, Fukui Prefecture)
 1960 Chile Earthquake


 Useful for studies of
   ・ Interior of the Earth
   ・ Largest earthquakes
Toroidal and Spheroidal Modes




Toroidal
                            Spheroidal


                        Dahlen and Tromp Fig. 8.5, 8.17
                Natural Vibrations of the Earth
Indexes describe spherical harmonics




                                         Shearer Ch.8.6
                                         Lay and Wallace, Ch. 4.6
Free Oscillations     l=1 m=1




          Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html
Free Oscillations    l=1 m=2




          Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html
Free Oscillations    l=1 m=3




          Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html
Structure: Free Surface


Earth is a not homogenous whole-space


Free surface causes many complications

   - surface waves
   - reflections (pP, sP, sS)
Summary
 Rays
    Velocity structure includes gradients, discontinuities
    and LVZ’s, causing complicated ray paths
      through the Earth (P, PKP, PcP)

 Wave theory explains
    ・ P and S waves
    ・ Static displacements
    ・ Surface waves

 Normal Modes
    The Earth rings like a bell at long periods
Why are observed seismograms so
            messy ?

								
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